Non-Calculator Square Roots

Up to 1960, and the Computer Age, if you wanted to find square roots at all accurately, you either used a mind-numbing ‘guess-and-check’ long multiplication approach – or an ingenious method attributed to Newton in about 1670. You’re going to use this last bit of Fraction Wizardry …

Step 1

To find the square root of, say, 5, we make a first guess – let’s call it  a1 = 2.

We know this is an under-estimate, but that’s no problem

Step 2

Since a square of area 5 has the same area as a  rectangle of area 5 (not the genius part !) , we can use  a1 and  b1  as ‘lower’ and ‘upper’ estimates for

We deduce the value of  b1 from the fact that  a1 x  b1 = 5, so

Step 3

Since  a1 is an under-estimate, and  b1 is an over-estimate, we take their average as our new, improved estimate for

So,

# The Problem :

A         Your first task is to repeat this procedure ( using  a2 instead of  a1 ) to find an even better estimate of   - I think you will be surprised how good it is after even just these 2 repetitions.

B         Then try the same method, from scratch, to find a good estimate of the value of

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